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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On real bisectional curvature for Hermitian manifolds

Authors: Xiaokui Yang and Fangyang Zheng
Journal: Trans. Amer. Math. Soc. 371 (2019), 2703-2718
MSC (2010): Primary 32Q05, 53C55
Published electronically: July 6, 2018
MathSciNet review: 3896094
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Abstract: Motivated by the recent work of Wu and Yau on the ampleness of a canonical line bundle for projective manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature $ H$, and when the metric is non-Kähler, it is slightly stronger than $ H$. We classify compact Hermitian manifolds with constant nonzero real bisectional curvature, and also slightly extend Wu and Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu and Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

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Xiaokui Yang
Affiliation: Morningside Center of Mathematics, Institute of Mathematics, HCMS, CEMS, NCNIS, HLM, UCAS, Academy of Mathematics; and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Fangyang Zheng
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210; and Zhejiang Normal University, Jinhua 321004, Zhejiang, China

Received by editor(s): October 23, 2016
Received by editor(s) in revised form: March 10, 2017, and October 24, 2017
Published electronically: July 6, 2018
Additional Notes: The first author was supported by China’s Recruitment Program and NSFC 11688101
The second author was supported by a Simons Collaboration Grant from the Simons Foundation.
Article copyright: © Copyright 2018 American Mathematical Society